Download How is Light Made?

How is Light Made?
Deducing Temperatures and
Luminosities of Stars
(and other objects…)
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Review: Electromagnetic (EM) Radiation
• EM radiation: regularly varying electric & magnetic fields
– can transport energy over vast distances.
• “Wave-Particle Duality” of EM radiation:
– Can be considered as EITHER particles (photons) or as waves
• Depends on how it is measured
• Includes all of “classes” of light
– ONLY distinction between X-rays and radio waves is wavelength λ
Increasing energy
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UV
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X Ult
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10-15 m
10-9 m
10-6 m
10-4 m
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10-2 m
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103 m
Increasing wavelength
Electromagnetic radiation is everywhere around us. It is the light that we see,
it is the heat that we feel, it is the UV rays that gives us sunburn, and it is the
radio waves that transmit signals for radio and TVs.
EM radiation can propagate through vacuum since it doesn’t need any medium
to travel in, unlike sound. The speed of light through vacuum is constant
through out the universe, and is measured at 3x108 meters per second, fast
enough to circle around the earth 7.5 times in 1 second. Its properties
demonstrate both wave-like nature (like interference) and particle-like nature
(like photo-electric effect.)
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Electromagnetic Fields
Direction
of “Travel”
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Sinusoidal Fields
• BOTH the electric field E and the magnetic
field B have “sinusoidal” shape
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Wavelength λ
λ
z Distance between two identical points on wave
One way of describing light is by its wavelength. Wavelength is the distance
between the two identical points on the wave. The wave must be steady (no
change in the oscillation and no change in its velocity) for it be possible to
measure the wavelength.
Wavelength is also shorthanded to a Greek letter Lambda.
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Frequency ν
time
1 unit of time
(e.g., 1 second)
z number of wave cycles per unit time
registered at given point in space
z inversely proportional to wavelength
The same exact wave can be described using its frequency. Frequency is
defined as the number of cycles of the waves per unit time. In the case shown,
the frequency would be 1.5, since there are exactly one and a half complete
cycles of the wave in the given time.
The frequency is inversely proportional to its wavelength.
Frequency is denoted by Greek letter “nu”.
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Wavelength and Frequency
λ = v/ν = c /ν (in vacuum)
z Proportional to Velocity v
z Inversely proportional to temporal frequency ν
z Example:
z AM radio wave at ν = 1000 kHz = 106 Hz
z λ = c/ν = 3 × 108 m/s / 106 Hz = 300 m
z λ for AM radio is long because frequency is small
Wavelength and frequency are related to one another by the wave’s velocity.
Wavelength is proportional (wavelength increases if velocity increases), and
wavelength is inversely proportional to frequency (wavelength decreases if
frequency increases).
An AM radio wave has a large wavelength, so it has a low frequency
(compared to other EM radiation.)
In the case of EM radiation, the velocity is the speed of light, denoted by c. the
speed of light is as mentioned before, approximately 3x108 meters per second.
Using algebra, one can solve for any one of the variables.
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“Units” of Frequency
⎡ meters ⎤
c⎢
⎣ second ⎥⎦ = ν ⎡ cycles ⎤
⎢⎣ second ⎥⎦
⎡ meters ⎤
λ⎢
⎥
cycle
⎣
⎦
⎡ cycle ⎤
= 1 "Hertz" (Hz)
1⎢
⎥
⎣ second ⎦
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Light as a Particle: Photons
z Photons: little “packets” of energy
z Energy is proportional to frequency
E = hν
Energy = (Planck’s constant) × (frequency of photon)
h ≈ 6.625 × 10-34 Joule-seconds = 6.625 × 10-27 Erg-seconds
Now the particle nature of EM radiation. These little packets of light is known
as photons. These photons carry a certain energy which is related to its
frequency. This energy is equal to Planck’s constant (h) multiplied by the
frequency of the photon. By substituting “nu” with the equation in the previous
slide, we can get the equivalent equation in terms of wavelength.
Planck’s constant is 6.6261 x 10-34 joule second
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Generating Light
• Light is generated by converting one class
of energy to electromagnetic energy
– Heat
– Explosions
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Converting Heat to Light
The Planck Function
• Every opaque object (a human, a planet, a star)
radiates a characteristic spectrum of EM radiation
– Spectrum: Distribution of intensity as function of wavelength
– Distribution depends only on object’s temperature T
• Blackbody radiation
ultraviolet
visible
infrared
radio
Intensity
(W/m2)
0.1
1.0
10
100
1000
10000
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Planck’s Radiation Law
• Wavelength of MAXIMUM emission λmax
is characteristic of temperature T
• Wavelength λmax ↓ as T ↑
As T ↑, λmax ↓
λmax
http://scienceworld.wolfram.com/physics/PlanckLaw.html
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Sidebar: The Actual Equation
B (T ) =
2hc 2
λ5
1
hc
e
λ kT
−1
• Derived in Solid State Physics
• Complicated!!!! (and you don’t need to know it!)
h = Planck’s constant = 6.63 ×10-34 Joule - seconds
k = Boltzmann’s constant = 1.38 ×10-23 Joules per Kelvin
c = velocity of light = 3 ×10+8 meter - second-1
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Temperature dependence
of blackbody radiation
•
As object’s temperature T increases:
1.
2.
Wavelength of maximum of blackbody spectrum (Planck
function) becomes shorter (photons have higher energies)
Each unit surface area of object emits more energy (more
photons) at all wavelengths
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Shape of Planck Curve
http://csep10.phys.utk.edu/guidry/java/planck/planck.html
• “Normalized” Planck curve for T = 5700K
– Maximum Intensity set to 1
• Note that maximum intensity occurs in visible region of
spectrum for T = 5700K
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Planck Curve for T = 7000-K
http://csep10.phys.utk.edu/guidry/java/planck/planck.html
• This graph is also “normalized” to 1 at maximum
• Maximum intensity occurs at shorter wavelength λ
– boundary of ultraviolet (UV) and visible
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Two Planck Functions
Displayed on Logarithmic Scale
http://csep10.phys.utk.edu/guidry/java/planck/planck.html
• Graphs for T = 5700K and 7000K displayed on
same logarithmic scale without normalizing
– Note that curve for T = 7000K is “higher” and its peak is
farther “to the left”
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Features of Graph of Planck Law
T1 < T2 (e.g., T1 = 5700K, T2 = 7000K)
• Maximum of curve for higher temperature
occurs at SHORTER wavelength λ:
– λmax(T = T1) > λmax(T = T2) if T1 < T2
• Curve for higher temperature is higher at ALL
WAVELENGTHS λ
⇒ More light emitted at all λ if T is larger
– Not apparent from normalized curves, must
examine “unnormalized” curves, usually on
logarithmic scale
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Wavelength of Maximum Emission
Wien’s Displacement Law
• Obtained by evaluating derivative of Planck
Law over temperature T
2.898 ×10−3
λmax [ meters] =
T [K]
Human vision range
400 nm = 0.4 µm ≤ λ ≤ 700 nm = 0.7 µm
(1 µm = 10-6 m)
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Colors of Stars
• Star “Color” is related to temperature
– If star’s temperature is T = 5000K, the wavelength
of the maximum of the spectrum is:
λ max
2.898 × 10 −3
=
m ≅ 0.579 µm = 579nm
5000
(in the visible region of the spectrum, green)
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Colors of Stars
• If T << 5000 K (say, 2000 K), the wavelength of
the maximum of the spectrum is:
λ max
2.898 × 10 −3
=
m ≅ 0.966 µm ≅ 966nm
3000
(in the “near infrared” region of the spectrum)
• The visible light from this star appears “reddish”
– Why?
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Blackbody Curve for T=3000K
• In visible region, more light at long λ
⇒ Visible light from star with T=3000K appears “reddish”
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Colors of Stars
• If T << 5000 K (say, 2000 K), the wavelength of
the maximum of the spectrum is:
λ max
2.898 × 10 −3
=
m ≅ 1.449µm ≅ 1450nm
2000
(peaks in the “near infrared” region of the spectrum)
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Colors of Stars
• Color of star indicates its temperature
– If star is much cooler than 5,000K, the
maximum of its spectrum is in the infrared and
the star looks “reddish”
• It gives off more red light than blue light
– If star is much hotter than 15,000K, its spectrum
peaks in the UV, and it looks “bluish”
• It gives off more blue light than red light
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Betelguese and Rigel in Orion
Betelgeuse: 3,000 K
(a red supergiant)
Rigel: 30,000 K
(a blue supergiant)
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Planck Curves for
Rigel and Betelgeuse
RIgel
Betelgeuse
Plotted on Log-Log Scale to “compress” range of data
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Luminosities of stars
• Sum of all light emitted over all wavelengths
is the luminosity
– A measure of “power” (watts)
– Measures the intrinsic brightness instead of
apparent brightness that we see from Earth
• Hotter stars emit more light at all wavelengths
through each unit area of its surface
– luminosity is proportional to T4 ⇒ small increase in
temperature makes a big increase in luminosity
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Luminosities of stars
• Stefan-Boltzmann Law
L = σT 4
L = Power emitted per unit surface area
σ = Stefan-Boltzmann Constant
≈ 5.67 × 10-8 Watts / (m2 K4)
• Obtained by integrating Planck’s Law over λ
• Luminosity is proportional to T4
⇒ small increase in temperature produces big
increase in luminosity
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Consider 2 stars with same
diameter and different T’s
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What about large & small stars
with same temperature T?
• Surface Area of Sphere ∝ R2
– R is radius of star
• Two stars with same T, different
luminosities
– the more luminous star must be larger to
emit more light
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How do we know that Betelgeuse
is much, much bigger than Rigel?
• Rigel is about 10 times hotter than
Betelgeuse (T = 30,000K vs. 3,000K)
– Rigel gives off 104 (=10,000) times more
energy per unit surface area than Betelgeuse
• But these two stars have approximately
equal total luminosity
– therefore diameter of Betelguese must be
about 102 = 100 times larger than Rigel
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So far we haven’t considered
stellar distances...
• Two otherwise identical stars (same
radius, same temperature ⇒ same
luminosity) will still appear vastly different
in brightness if at different distances from
Earth
• Reason: intensity of light inversely
proportional to the square of the distance
the light has to travel
– Light wave fronts from point sources are like
the surfaces of expanding spheres
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Use Stellar Brightness Difference
• If one can somehow determine that brightnesses
of 2 stars are identical, then use their relative
brightnesses to find their relative distances
• Example: the Sun and α Cen (alpha Centauri)
– spectra look very similar
⇒ temperatures are almost identical (from Planck
function)
• diameters are also almost equal
• deduced by other methods
⇒ luminosities about equal
• difference in apparent magnitudes ⇒ difference
in relative distance
– Check using parallax distance to α Cen
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The Hertzsprung-Russell Diagram
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Hertzsprung-Russell (“H-R”) Diagram
• Graphical Plot of Intrinsic
Brightness as function of Surface
Temperature
• 1911 by Hertzsprung (Dane)
• 1913 by Henry Norris Russell
• Stars Tend to “Cluster” in Certain
Regions of Plot
– “Main Sequence”
– “Red Giants” and “Supergiants”
– “White Dwarfs”
• Star “Types” based on
Temperature
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Star Types
O B
A
F
G K M
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